Order parameter ferroelectricity

Continuous models. With discovery of antiferroelectrics a question has arised about the possible structures and order parameters describing the new phases. Since all the structures are based on the single tilted SmC layers of the same C2 symmetry, it was suggested to use the same c-director to characterised each pair of neighbour layers (bilayer model [28]). Taking two neighbour layers i and i + 1, one writes two order parameters, ferroelectric and antiferroelectric, both in terms of the director components nyii ), see Eqs. (13.9a, b) where z is the normal to... 

At high temperatures, ferroelectric materials transform to the paraelectric state (where dipoles are randomly oriented), ferromagnetic materials to the paramagnetic state, and ferroelastic materials to the twin-free normal state. The transitions are characterized through order parameters (Rao Rao, 1978). These order parameters are characteristic properties parametrized in such a way that the resulting quantity is unity for the ferroic state at a temperature sufficiently below the transition temperature, and is zero in the nonferroic phase beyond the transition temperature. Polarization, magnetization and strain are the proper order parameters for the ferroelectric. 

Ferroelectric-paraelectric transitions can be understood on the basis of the Landau-Devonshire theory using polarization as an order parameter (Rao Rao, 1978). Xhe ordered ferroelectric phase has a lower symmetry, belonging to one of the subgroups of the high-symmetry disordered paraelectric phase. Xhe exact structure to which the paraelectric phase transforms is, however, determined by energy considerations. 

It is useful to check whether this kind of relations is valid for other systems like ferromagnetics and ferroelectrics too. Here the order parameters are the magnetization M and the polarization P, respectively. At high temperatures (T > Tc), and zero external field these values are M = 0 (paramagnetic phase) and P = 0 (paraelectric phase) respectively. At lower temperatures close to the phase transition point, however, spontaneous magnetization and polarization arise following both the algebraic law M, P oc (Tc - Tf. 

In addition to lowering V th, ferroelectric nanoparticles such as BaTi03 or Sn2P2S6 [144, 156, 318-323] have also been shown to increase the nematic-to-isotropic phase transition temperature (TN/Iso) and the order parameter of the nematic host [142, 320, 324-326], which are thought to have their origin in a coupling of the electric dipole moment of the particles with the orientational order of the surrounding nematic molecules (Fig. 6). 

Key Words Dipolar glasses, Ferroelectric relaxors, Conducting polymers, NMR line shape, Disorder, Local polarization related to the line shape, Symmetric/asymmetric quadrupole-perturbed NMR, H-bonded systems, Spin-lattice relaxation, Edwards-Anderson order parameter, Dimensionality of conduction, Proton, Deuteron tunnelling. 

A number of studies have treated the effects of impurities on phase transitions from a theoretical perspective (Halperin and Varma 1976, Hock et al. 1979, Levanyuk et al. 1979, Weyrich and Siems 1981, Lebedev et al. 1983, Bulenda et al. 1996, ScWabl and Tauber 1996). By and large, however, theoreticians have focused on the way in which local interactions between defect fields and the order parameter produce an anomalous central peak in neutron scattering cross-sections of impure ferroelectrics up to 65°C above the critical temperature (Shirane and Axe 1971, Shapiro et al. 1972, Muller 1979). 

This argiimcntation is further supported by the predictions of a simple mean-field theory of the ferroelectric transition, which was originally presented in Ref. 257. Within this theory, we neglect any stratification (i.e., inhomogeneities of the local density) as well as any oscillations in the order parameter (which are indeed observed in the computer simulations). We also neglect nontrivial interparticle correlations. Our system can then be viewed as a system composed of N uncorrelated dipolar particles individually interacting with the mean field... 

Once the probabilities are known, other physical quantities, which are function of the occupation probabilities, can be calculated from (A) — J2yPy y- or order parameters for order-disorder phase transitions. Different examples will appear in the following. For instance, the orientational contribution to the absolute polarization of the ferroelectric compound pyridinium tetrafluoroborate was estimated from 2H NMR temperature-dependent measurements on the perdeuterated pyridinium cations.116 The pyridinium cation evolves around a pseudo C6 axis, and the occupation probabilities of the different potential wells were deduced from the study of 2H NMR powder spectra at different temperatures. The same orientational probabilities can be used to estimate the thermodynamical properties, which depend on the orientational order of the cation. Using a generalized van t Hoff relationship, the orientational enthalpy changes were calculated and compared with differential scanning calorimetry (DSC) measurements.116... 

Fig. 8. Simple model of order-disorder or displacive ferroelectric phase transition. Left, ferroelectricity by relative displacement of the anion and cation sublattices (a) displacive model, where r — 0 in the HTP and the atoms are translated by r/0 in the LTP. The order parameter is r. (b) Order-disorder model in the high-temperature phase, the ions are symmetrically disordered with equal probabilities p+ — p — 1/2 over two positions r — +rQ. In the low-temperature phase, the occupancies of the sites become unequal with probabilities p p +. The order parameter is the difference Ap — p+—p. The spontaneous polarization Psocr and PsccAp for the displacive model and order-disorder model, respectively. Right, ferroelectricity by alignment of molecular dipoles (c) displacive model in the HTP, all the molecules are aligned with a = 0 in the LTP, the molecules are rotated around the center of inversion with angles +a/0, the order parameter is a. (d) Order-disorder model. The spontaneous polarization Ppx ct and PsccAp for the displacive model and order-disorder model, respectively.

Ferroelectric liquid crystal 2H, SAXS, order parameter 241... 

The flexoelectric effect is a phenomenon where a space variation of the order parameter induces polarization. Chiral polar smectics are liquid crystals formed of chiral molecules and organized in layers. All phases in tilted chiral polar smectic liquid crystals have modulated structures and they are therefore good candidates for exhibiting the flexoelectric effect. The flexoelectric effect is less pronounced in the ferroelectric SmC phase and in the antiferroelectric SmC. The flexoelectric effect is more pronounced in more complex phases the three-layer SmCpu phase, the four-layer SmCFi2 phase and the six-layer SmCe a phase. 

It is known that the crystal symmetry defines point symmetry group of any macroscopic physical property, and this symmetry cannot be lower than corresponding point symmetry of a whole crystal. The simplest example is the spontaneous electric polarization that cannot exist in centrosymmetric lattice as the symmetry elements of polarization vector have no operation of inversion. We remind that inversion operation means that a system remains intact when coordinates x, y, z are substituted by —x, —y, —z. If the inversion center is lost under the phase transition in a ferroic at T < 7), Tc is the temperature of ferroelectric phase transition or, equivalently, the Curie temperature), the appearance of spontaneous electrical polarization is allowed. Spontaneous polarization P named order parameter appears smoothly... 

Ferroelastics are the materials where the order parameter is mechanical deformation that spontaneously appears at transition temperature. As it was mentioned in Chap. 1, Aizu [74, 75] was the first who gave the formal definition of ferroelasticity as the property that can exist by itself in the materials which are neither ferroelectrics nor ferromagnets. 

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